Planarity of a spanning subgraph of the intersection graph of ideals of a commutative ring I, nonquasilocal case

The rings considered in this article are nonzero commutative with identity which are not fields. Let $R$ be a ring. We denote the collection of all proper ideals of $R$ by $\mathbb{I}(R)$ and the collection $\mathbb{I}(R)\setminus \{(0)\}$ by $\mathbb{I}(R)^{*}$. Recall that the intersection graph of ideals of $R$, denoted by $G(R)$, is an undirected graph whose vertex set is $\mathbb{I}(R)^{*}$ and distinct vertices $I, J$ are adjacent if and only if $I\cap J\neq (0)$. In this article, we consider a subgraph of $G(R)$, denoted by $H(R)$, whose vertex set is $\mathbb{I}(R)^{*}$ and distinct vertices $I, J$ are adjacent in $H(R)$ if and only if $IJ\neq (0)$. The purpose of this article is to characterize rings $R$ with at least two maximal ideals such that $H(R)$ is planar.